FUNCTION. 
in which the fine and es is éxpreffed in terms of i 
ginary exponentials, which may be seated as one of t 
nioft curious analytical dibown-s of the laft century 
Thefe formule may likewife he derived immediately from 
the oo gee! gi 
+ Xy Ce n this manner that Euler has given them in 
vol. vii. of the Mifcellanea einen fia. 
he expreffion the value of arcs in imaginary logarithms 
was difcovered b n Bernou illi, we as at it in sree 
d ies which expreffes the 
by the tangent, by integrating the fame element in 
"The equation one x + fin. x Soy zetv- 
where the radical + may equally have the fign + 
forms the bafis of the theory of the calculation | of an 
gles; for by multiplying this equation by the fimilar equa: 
tion cof. y + fin yW— § = e7V¥~! 
(cof x + fine “—1) (cof. y ye 
e (+ P/—1, and fubltituting in the fame equation x + y in 
A place of x, 
3 
cof (s+ 9) + fin. (wy) Wma = etn yay 
uae Ae expanding the, aha . 
fin. (cof.x. fin. y + cof. y 
fin.. x) /—1= cof. (x es + fin. (x + y) yeas 
and as this equation ~ fubfift for the two figns of “/ — 
"a follows.that fepara’ 
= a . + J) 
= J} 
which formulz may be demonitrated pou ily, and form 
a‘ bafis of the pel of angles. 
addition the following inveftigation. of what is 
fall called the — method of fluxions, as given by Mr. 
Woodhoufe, er is referred to the article Econ, 
the fubftance o ete is likewife taken from the works of 
this learned an 
In sont adit acaba to the direét operations of multipli- 
cation and pea epee and evolution are called reverfe 
pee erations 3. but w .the refults-of thefe operations are 
ehended padre acominen formula (9 + x. Ae 
c.), or exprefled by a general pcre it is commodious 
to confider the latter method by which algebr aic finions 
are expanded, and any term ae . a -— method; and 
the method by which, fro of an expanded 
fundion, we afcend to the ee | eons a reverfe me- 
hod. 
Onthe former method depends the d habia mae by 
which from ¢@ x the fecond, third, &c. of 9 (a+ dx 
expanded, are affigned ; on - reverfe depend the Sere) 
7 Dy which from X. oe oe 
which ‘FinGion ae! 
Avs en “Tecond ania tals, ie or are oom te aie third, 
&c. termsof f (x + dx), f' (« +d.) &c..expa anded, 
are to be affigned. Theinterea ae is unable, mee 
in all cafes, to affign the original or primitive functions of 
from which a differential Ade : derived ; its rules 
thofe for the extraction of roots) are eftabli ie 
_ {peétion of the direc oe ne which, from primitive 
functions, differentials are obtain d 
= mem dv; .*, from ma ”—~*. da -we may afcend to 
x” by this proccfs; increafe the index (m of th 
power of the aes quantity () in the differential 
2 
ven which (expr refs the functions. fin. x, - 
(ma"—-*. da) by 15 and then divide the diff t 
- index fo a . m), and by the ‘cme ne 
f this procefs , toe 
(x"). ‘ bet 
which - integral is hae 1, the 
tes),. 
then d-? renee mat ord' a (2 = 4”, 
Pmt 
Again, d=! re dw) == 3 andd~? (ar = al 
Sia rida 
Again, from what has 
receded it 
fun¢tion of x pene p= D that if p bea 
the differential of @ p @p.dpor 
= sce dp; hence to find the integral or primitive 
function of fuch a differential as Do h 
to find the oo of which D @ Hae "lifferentad 
ve obferved, that the integral of a 
q a fi @ion can cn] e 
und under ere a for if F pbe the fun@ion 
of which fp is the differential co-efficient, then fp = F ea 
dp 
dF 
“Sfp. fee ge -ifdp=adg, a an inva- 
riable eal the nee can be found: but if d p 
does = dg, the ntegral cannot be found, at leaft not 
ianetetle foe inane, le g= (a i Ce ae 
(3 Peta dae np=atba + ¢ «*, and ..d 
éd p56 es ied bdu+te2 2cad., confe- 
quently the integral can be found = ce tbe Sea Sd acd ale 
am I 
Le let Sf eee d So 
1 dx, and xd x, co 
et ‘fe integral - ee aa at b ied not imme- 
diately) except r = r except the index of the 
power of the erable arsneey, oe the vinculum, be lefs 
by unity thanthe index of the power of the variable quan- 
tity under- the vinculum ; 
again, if fp. dg= (a+ ba" 4 cx ig 
(nbat +rex')da, ~p= athe” tex" and. dp = 
nhat-*da+reea,-'da; but d gq=nba'du + 
z'd x, confequen ntly the integral of d t 
nd (at leaft not aamediat ly). ¢ lof ibe = pi ae ae 
=r—t; again, if fp.g =e". eda, p=x, fp 
e, dp= 2xda, butd gauwda, confequently the in- 
7) 
2 
’ ee and are ob- 
(a + 5 x), van eer 
fe. 
— 
— 
tegral of e, wd x-can be found ( = 
.dx, are — 
The integrals of a*.d x, e a 
tained on the Ly ars as the integrals of x"—! d x, 
r hat is, b ection ce 
the ie "vhich te operation of differentiation performed 
on known funétions, 
a+bx 
a al + bx)= X» 
Hence, reverfely, d-? (6d x) =a+ bx, or'= a! + 3 xy 
= qit bx, or = &e. 
fe d(ait+bx)s=bdx 
= bas, d (dt bs) = 0d 
Thefe conftant quantities a, a’, a", &c. are called correc. 
tions, and j in integrating fuch an -expreffion as 6 d x, which 
of the corrections a, a’, a" is to be ufed, muft be determined 
by 
